A272170 Second most significant bit of Fibonacci numbers > 1 written in base 2.

0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0

Offset: 3

Author

Andres Cicuttin, Apr 21 2016

Keywords

• nonn
• base

Comments

It is conjectured that there are no more than two consecutive "0's" or “1’s” (tested up to n=10^5). The sequence looks quasiperiodic and its Fourier spectrum seems to have a fractal structure.

Examples

			(second MSB in parenthesis)
  n   A000045(n)      A004685(n)
  3      2       ->   1(0)
  4      3       ->   1(1)
  5      5       ->   1(0)1
  6      8       ->   1(0)00
  7      13      ->   1(1)01
  8      21      ->   1(0)101
  9      34      ->   1(0)0010
  10     55      ->   1(1)0111
...
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A000045, A004685, A079944, A271591.

Programs

Mathematica

nmax = 120; Table[IntegerDigits[Fibonacci[j], 2][[2]], {j, 3, nmax}]

PARI

a(n) = binary(fibonacci(n))[2]; \\ Michel Marcus, Apr 25 2016

Python

A272170_list, a, b = [], 1 ,1
for n in range(3,10001):
    a, b = b, a+b
    A272170_list.append(int(bin(b)[3])) # Chai Wah Wu, Feb 07 2018

Formula

a(n) = floor(A000045(n)/(2^(ceiling(log_2(A000045(n) + 1)) - 2))) - 2.

a(n) = A079944(A000045(n)-2). - Michel Marcus, Apr 22 2016